Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban–Lions domains with holes

AbstractIn this article, we consider the following eigenvalue problems(∗)λ−Δu+u=λ(f(u)+h(x))inΩ,u>0inΩ,u∈H01(Ω), where λ>0, N⩾2 and Ω⊂RN is the upper semi-strip domain with a hole in RN. Under some suitable conditions on f and h, we show that there exists a positive constant λ∗ such that Eq. (∗)λ has at least two solutions if λ∈(0,λ∗), a unique positive solution if λ=λ∗, and no positive solution if λ>λ∗. We also obtain some further properties of the positive solutions of (∗)λ